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Markov property of point processes
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  • Published: September 1987

Markov property of point processes

  • Hans G. Kellerer1 

Probability Theory and Related Fields volume 76, pages 71–80 (1987)Cite this article

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Summary

A point process \(\mathcal{N}\) on R + can be represented by the associated counting process (ξ t ;t∈ R +) or by the associated sequence of jump times (τ n ;n∈ Z +) and in accordance may possess two types of Markov property. The present paper first clarifies their mutual dependence, leading in particular to the notion of “weak multiplicativity” for the joint distribution of two consecutive jump times. Then, by means of results from a previous paper, a uniquely determined “Markov variant” \(\tilde mathcal{N}\) is assigned to \(\mathcal{N}\) without changing the one-dimensional marginals. This provides in particular a new characterization of the Poisson process by these marginals and the adequate Markov property. Further applications concern the explicit construction of the compensator and certain transition probabilities of \(\tilde mathcal{N}\).

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References

  1. Bremaud, P.: Point processes and queues. Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  2. Jacod, J.: Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 235–253 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kallenberg, O.: Characterization and convergence of random measures and point processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 27, 9–21 (1973)

    MATH  MathSciNet  Google Scholar 

  4. Kellerer, H.G.: Order conditioned independence of real random variables. Math. Ann. 273, 507–528 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Rényi, A.: Remarks on the Poisson process. Studia Sci. Math. Hungar. 2, 119–123 (1967)

    MATH  MathSciNet  Google Scholar 

  6. Szász, D.: Once more on the Poisson process. Studia Sci. Math. Hungar. 5, 441–444 (1970)

    MathSciNet  Google Scholar 

  7. Yushkevich, A.: On strong Markov processes. Theory Probab. Appl. 2, 181–205 (1957)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstrasse 39, D-8000, München, Federal Republic of Germany

    Hans G. Kellerer

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  1. Hans G. Kellerer
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Cite this article

Kellerer, H.G. Markov property of point processes. Probab. Th. Rel. Fields 76, 71–80 (1987). https://doi.org/10.1007/BF00390276

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  • Received: 03 April 1986

  • Revised: 05 April 1987

  • Issue Date: September 1987

  • DOI: https://doi.org/10.1007/BF00390276

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Keywords

  • Stochastic Process
  • Probability Theory
  • Poisson Process
  • Mathematical Biology
  • Point Process
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